Let $D_N$ be the $N$th Dirichlet kernel, $D_N = \sum_{k = -N}^N w^k$, where $w = e^{ix}$. Define the Fejer kernel to be $F_N = \frac{1}{N}\sum_{k = 0}^{N-1}D_k$. Then $$F_N = \frac{1}{N}\frac{\sin^2(N x/2)}{\sin^2(x/2)}$$.
So far I have $D_k = \frac{w^{k+1} - w^{-k}}{w-1}$, and so $$ \begin{align*} F_N &= \frac{1}{N}\sum_{k=0}^{N-1} D_k \\ &= \frac{1}{N(w-1)}\sum_{k=0}^{N-1} (w^{k+1} - w^{-k}) \\ &= \frac{1}{N(w-1)}\left ( w\sum_{k=0}^{N-1} w^k - \sum_{k=-N+1}^0 w^k \right ) \\ &= \frac{1}{N(w-1)}\left ( \frac{w(w^N - 1)}{w-1} - \frac{1-w^{-N + 1}}{w-1} \right ) \\ &= \frac{1}{N(w-1)^2}\left ( w^{N+1} +w^{-N + 1} - (w + 1) \right ) \end{align*} $$